Despite the persistence of some sensory processing during sleep, it is unclear whether high-level cognitive processes such as speech parsing are also preserved. We used a novel Concurrent Hierarchical Tracking (CHT) approach for studying the depth of speech processing across wakefulness and sleep while tracking neuronal activity with EEG. We found that responses to the auditory sound stream remained intact; however, the sleeping brain did not show signs of hierarchical parsing of the continuous stream of syllables into words, phrases, and sentences. The results suggest that sleep imposes a functional barrier between basic sensory processing and high-level cognitive processing. This paradigm also holds promise for studying residual cognitive abilities in a wide array of unresponsive states.